Multiphase flow modeling and numerical simulation of pyroclastic density currents

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Universit`

a di Pisa

Facolt`

a di Scienze Matematiche, Fisiche e Naturali

School of Graduate Studies “Galileo Galilei”

Ph.D Thesis in Applied Physics

Multiphase flow modeling and

numerical simulation of

pyroclastic density currents

Author:

Dr. Simone Orsucci

Advisors: Dr. Tomaso Esposti Ongaro1 Prof. Fulvio Cornolti2

1_{Istituto Nazionale di Geofisica e Vulcanologia, sezione di Pisa} 2 _Universit`_{a di Pisa, Dipartimento di Fisica ”E.Fermi”}

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List of Tables

1 Transport equations . . . 61

2 Changes in closure equations . . . 64

3 Input particle diameter and initial volume fraction for the nu-merical simulations of the dam-break problem . . . 82

List of Figures

1 Sketch of magma ascent from chamber to vent in an explosive eruption . . . 11

2 Buoyant plume of ash rises at Kirishima Volcano, Japan . . . 12

3 Pyroclastic flows descend the south-eastern flank of Mayon Volcano, Philippines . . . 13

4 Sketch of a structure of an incompressible turbulent PDC . . . 15

5 Spatial distribution of pyroclasts into the atmosphere . . . 18

6 Dependency of the effective solid viscosity on the solid volu-metric fraction . . . 30

7 Sketch of particle-particle collision . . . 34

8 Granular temperature ratio for granular mixture . . . 56

9 Typical 2D Cartesian reference volumes . . . 67

10 Sketch of the computational stencil needed to compute FV fluxes in 2D . . . 71

11 PDAC main flow-chart . . . 74

12 Sketch of the iterative bisecant method . . . 77

13 Current with dp = 10µm and s,0 = 5 × 10−4 . . . 84

14 Sketch of a pyroclastic density current structure . . . 84

15 Particle volume fraction in the current front and in the current body for dp = 10µm . . . 85

16 Sedimenting currents . . . 86

17 Particle volume fraction in the current head and in the current body for sedimenting currents . . . 87

18 Particle volumetric fractions for different particle sizes . . . 88

19 Front position for different diameters . . . 90

20 Sketch of the geometric assumptions of the box model. . . 91

21 Linear fit of l3/2_{(t) for monodisperse particle-laden . . . .} ₉₅

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23 Non-dimensional front position for sedimenting currents . . . . 96

24 linear fit of l3/2(t) for different values of g0 . . . 98

25 Gravity current with buoyant interstitial fluid (T = 700K) . . 100

26 Gravity current with buoyant interstitial fluid at fixed time . . 101

27 Front current position and current volume of buoyant current with different particle diameters. . . 102

28 Front current position for different initial temperatures . . . . 105

29 Front position for different boundary conditions at the ground. 107 30 Front position for different rheological models . . . 108

31 Gravity currents obtained with different rheological models . . 109

32 Particle concentration at fixed time (a) and instantaneous par-ticle concentration at fixed position (b). . . 110

33 Solid volumetric fraction and granular temperature in the basal layer . . . 111

34 Total granular thermal energy during current propagation and contour of granular-energy density at time t = 100s . . . 113

35 Solid fraction for initial concentration s = 10% . . . 115

36 Horizontal velocity for initial concentration s = 10% . . . 116

37 Vertical profiles of granular temperature . . . 117

38 Vertical profiles solid sher viscosity . . . 118

39 Vertical profiles of granular pressure . . . 119

40 Sketch of energy line approach . . . 122

41 1D invasion model . . . 125

42 Invaded area at Campi Flegrei caldera for different current volumes and vent locations. . . 127

43 Sensitivity of the invasion model . . . 128

44 Workflow scheme for the hazard mapping . . . 130

45 Probability maps of new vent opening location in CF caldera 130 46 Maps of PDC invasion probabilities . . . 132

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List of Symbols

i Volume fraction occupied by phase i

vi Fluid velocity of phase i

Ti Temperature of phase i

hi Enthalpy of phase i

Pg Gas pressure

Ps Granular pressure of solid phase s

θ Granular temperature

Pi Fluid stress tensor of phase i

c Single particle velocity

ρg Gas density

ρs Bulk density of granular phase s

Di,j Interphase drag coefficient

µi Shear viscosity of phase i

µb,s Bulk viscosity of solid phase s

ki Thermal conductivity of phase i

λ Conductivity of granular heat

γ Dissipation rate of granular temperature

dp Particle diameter

mp Particle mass

ep Inelastic restitution coefficient

fs s-particle velocity distribution function

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Re Reynolds number

Fr Froude number

g0 Reduced gravity

h Height of the current in box model description

l Density current front position

ρc Mean density of gravity current

ρa Density of surrounding fluid

ρb Density of interstitial fluid

g0_p Particulate reduced gravity ws Settling velocity

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Acknowledgements

First and foremost I want to thank my advisor Dr. Tomaso Esposti Ongaro. It has been a honor to be his first Ph.D. student. He has supported me con-stantly, making a fundamental contribution to this thesis. Moreover, he has shown me what are the necessary qualities of a good researcher. Without his encouragement and constant guidance, I could not have completed my work. I am also thankful to Prof. Fulvio Cornolti, which is the second advisor of my thesis. He gave me several suggestions, ideas and guided me during these years of work. I would like to thank Dr. Augusto Neri, the former Directior ot the Pisa department of the Istituto Nazionale di Geofisica e Vulcanologia. He supported me and thanks to him I achieve a more global view of the studied problem. I thank Prof. Francesco Pegoraro, Director of the council of PhD course in Applied Physics. I learned a lot through our discussions, which have been a continuous source of reflection. I am grateful to the two re-viewers, Dr. Michele Larcher (Universit`a di Trento) and Prof. Piero Salatino (Universit`a di Napoli) for the time spent in revising the manuscript and for their thorough examination and useful suggestions which greatly improved the thesis.

I am thankful to the project SAFER (Services and Application for Emer-gency Response), promoted by the European Union, and the to the project DPC-V1 (on the probabilistic volcanic hazard assessment at Campi Flegrei caldera), promoted by the Dipartimento della Protezione Civile (italian civil protection) for their financial support.

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1 Introduction

Pyroclastic density currents are hot and high-density mixtures of gas and particles of different sizes which can be originated during explosive volcanic events by the fragmentation of a viscous magma. The dynamics of these currents are not fully understood and their complexity is deeply related with their multiphase nature, since the particle concentration influences some key aspect of current propagation (e.g, traveling speed, turbulent mixing with the atmosphere, sole erosion). Moreover, the pyroclastic density currents are among the most destructive phenomena associated with volcanic eruptions and so the problem is important not only from a pure scientific point of view, but has relevant consequences for the assessment of their related hazard.

In last decades these currents have been studied adopting a continuum description. In some of the proposed models a pyroclastic density current is modeled as a single homogeneous fluid, simplifying the effects of dispersed particles. A different approach treats the particles as different fluid phases, for which the transport equations are closed with semi-empirical relations, in particular for solid stress tensor and transport coefficients. An alternative approach, which has been developed to describe particulate systems, adopts the formalism of the kinetic theory and statistical mechanics to describe the interaction between solid particles instead of molecules. Such models are widely used in different disciplinary fields, especially in the engineering context, but their application to geophysics is not consolidated. A relevant part of this thesis is dedicated to a review of the derivation of the equations for solid phases using the kinetic theory of granular flows. Starting from the Liouville equation, we examine the effectiveness of the assumptions necessary to obtain the transport equation and which define the range of applicability of the model.

The main goal of this thesis is to investigate the effects of the particle concentration and stratification on current dynamics, by means of numerical simulations performed adopting a kinetic-theory-based model. We study the effect of a kinetic-based this model for solids on the current propagation and compare them with semi-empirical models. Finally, complex modeling tools are used to test and validate simplified simulation tools, not too computation-ally demanding, as requested for probabilistic studies of pyroclastic density hazard assessment in active volcanic regions. In this thesis we analyze an integral model for pyroclastic density currents, which considers the gravity current as homogeneous preudo-fluids in hydrostatic equilibrium whith

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ex-ternal ambient and does not consider the multiphase nature of particulate flows. Since it represents an over-simplification of gravity currents, we inves-tigate its range of applicability and calibrate it against numerical simulation obtained with the newly developed multiphase flow model, that describes the current dynamics with more accuracy and can predict non-equilibrium phenomena, such as gas-particle decoupling and current stratification, which can play a fundamental role on current propagation.

1.1 Mechanism of explosive eruptions

Explosive eruptions are the most powerful and destructive type of volcanic activity. They are characterized by ejection in the atmosphere of a mixture of gases and fragmented magma and lithics at high velocity, temperature and pressure [Sigurdsson et al., 1999]. This is opposed to effusive eruptions where the magma is erupted as a liquid melt with dispersed bubbles and crystals [Sigurdsson et al., 1999].

The complex mechanism leading to magma fragmentation is associated with the strongly non-linear rheological changes of the magma during its rise towards the surface, as associated with gas exsolution, bubble growth and crystallization. Fragmentation is affected also by other factor, as the geom-etry of the volcanic conduit and presence of phreatic water in surrounding material [Papale, 1999]. The ascending path of magma that originates an explosive eruption is sketched in Fig. 1.

The type and efficiency of the fragmentation process influences directly the explosivity associated with the eruptions. In general, the products re-leased into the atmosphere during explosive eruptions are: volcanic gases (mainly H2O and CO2), fine ash, expanded pumices, crystallized magma,

lithic fragments deriving from erosion of the conduit. Any volcanic frag-ment that was hurled through the air by volcanic activity are called as pyro-clasts, and the grain size distribution are characterized by a large variability, not only between two different explosive events, but also for a single event [Sparks, 1976]. E.g., the grain-size distribution of pyroclastic material es-timated in the Taupo eruption (occurred about 1,800 years ago, and that represents the most violent eruption in the world in the last 5000 years) cover the range from 30µm to about 50mm. In some other eruption also wider distributions are observed [Kaminski and Jaupart, 1998].

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Figure 1: Sketch of magma ascent from chamber to vent in an explosive eruption

The mixture of gas and particles ejected during a volcanic eruption is called eruption column, and different types of events are characterized by their particular eruption column structure and dynamics, depending on com-binations of discharge rate, magma density and viscosity, and the shape of the magma chamber and conduit. The gas-particle mixture can be shot out from the volcano with velocities up to several hundreds of meters per second, with a mean density that is higher than that of external atmosphere, due to the presence of transported solids. This ejected jet will mix with relatively cool air, that becomes heated and expands, causing a decrease of effective density of the rising column (also called plume). At the same time, friction between column boundaries and external atmosphere causes drag and con-sequent gravitational fall out of particles [Sparks, 1986, Sigurdsson et al., 1999]. An example of a volcanic plume is represented in Fig. 2.

When a rising eruption column does not incorporate enough air to reduce its density down to value of the external atmosphere, the rising jet deceler-ates until stagnating. Since the plume density when the column stops is still more dense with respect to the air, the particle-gas mixture start to fall

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Figure 2: Dense plume of ash rises at Kirishima Volcano, Japan. Photograph from Ky-odo/Reuters

down. When this occurs, the part of the column that gravitationally col-lapses, called tephra fountain, can originate a hot and high-density current, composed by a mixture of gas and pyroclasts. These currents, called Pyro-clastic Density Currents (PDCs), can travel horizontally from volcano flanks for several kilometers, destroying nearly everything along their path. Gravi-tational collapse of eruption column is favored for large vent radii, low vent velocities, high solid mass fractions at the vent and by large particle sizes [Woods, 1995, Valentine and Wohletz, 1989]. Fig. 3 shows the PDC origi-nated by explosive activity of Mayon Volcano, Philippines, during eruption of 15 September 1984.

1.2 Phenomenology of PDCs

Pyroclastic density currents can be originated also as a consequence of other explosive volcanic events. The most common mechanism is the collapse of a Plinian column, but common phenomena include the rapid expansion and de-compression (blast), of an eruptive pyroclastic mixture into the atmosphere, as a consequence of the explosion or collapse of a part of volcano. The

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Figure 3: Pyroclastic flows descend the south-eastern flank of Mayon Volcano, Philippines, during eruption of 15 September 1984. Maximum height of the eruption column was 15 km above sea level, and volcanic ash fell within about 50 km toward the west. There were no casualties because more than 73,000 people evacuated the danger zones as recommended by scientists of the Philippine Institute of Volcanology and Seismology. Photograph C. Newhall

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known 18 May 1980 Mt. St. Helens eruption is an excellent example of a directed blast from the side of a volcano. Another phenomenon that can generate a PDC is a hot avalanche of very concentrate pyroclastic material, consequent to the structural collapse of a lava dome [Sigurdsson et al., 1999]. PDCs can encompass a wide range of particle concentration, from py-roclastic surges, which are dilute turbulent suspensions (with a typical solid volumetric fraction less than 10−3), to pyroclastic flows, as are usually named the fluidized granular avalanches [Fagents et al., 2013] (for currents with a solid fraction of about 0.1).

These mixtures are non-uniform inside, with a high vertical stratifica-tion. As a first approximation it is possible to consider a PDC as com-posed by two main layers: an upper lighter part, and a basal denser part (in which the volumetric fraction can reach values close to the maximum packing fraction ∼ 0.7 − 0.8). These two regimes coexist in most PDCs [Valentine and Wohletz, 1989, Branney and Kokelaar, 2002]. The pyroclasts result from magma fragmentation and their granulometry commonly can vary from micron-sized ash to centimeter-sized and sometimes meter-sized volcanic bombs. PDCs have typical volumes from 106 m3 to 1012m3 in bigger events. These currents commonly travel over distances of several kilometers to several tens of kilometers, with propagation speed that can reach the value of about 100 m/s. The invaded area and the current deposit depend primar-ily on the total mass collapsed and on the excess density with respect to the atmospheric density.

The dynamics and propagation of PDCs depend on several factors, which interact in a complex way. The main of these is the excess of mixture den-sity with respect the atmospheric, that gives rise to the motion on the lower interface (volcano flanks and neighboring area). This density difference de-pends on the presence of particles in PDCs, and the evolution of particle concentration (and consequently current density) in time and space is not trivial, since it may change due to some, competing, effects. At the interface of the PDCs with air, turbulence produces a low-concentration mixing zone above the current body [Cushman-Roisin and Beckers, 2007]. When the flow becomes locally lighter than atmosphere we observe a rising column, due to the reversal of buoyant force, and the current may decelerate or stop [Sig-urdsson et al., 1999].

The solid concentration may change also by the effect of erosion and con-tinuous particle falling at the ground. The topography does not influence

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Figure 4: Sketch of a structure of an incompressible turbulent PDC [Branney and Kokelaar, 2002].

the dynamics only with soil erosion, since the natural obstacles may deviate and stop the flow. The lighter part of PDCs, called surges, are only weakly influenced by the topography, and generally overcomes the obstacles. On the other hand, the basal dense flows are more influenced by obstacles, and can be subject to variation of velocity magnitude, direction and vertical profile [Branney and Kokelaar, 2002, Fagents et al., 2013], and the final invaded area is generally highly related with the topography. The structure of an impulsive and dilute PDC is represented in Fig. 4. We can identify some main areas, subdividing the current into a head, a body and a tail.

Pyroclastic density currents can be classified also according to their dura-tion. Generally they are short-lived, since are formed by an impulsive event, and can be considered as highly unsteady flow. Clearly, all pyroclastic den-sity currents have finite duration and thus all are inherently unsteady, but pyroclastic fountaining eruptions may sustain pyroclastic density currents for periods up to several hours or more, during which we may identify periods of quasi-steady flow [Branney and Kokelaar, 2002].

PDCs are probably the most destructive phenomena associated with the volcanic activity. In particular their dangerousness is related with the high

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dynamic pressure (which gives the measure of kinetic energy on unit volume), high-temperature and deposit thickness (deposits from single flows range in volume from less than 0.1 km3 to over 3000 km3 [Sigurdsson et al., 1999]). The main values exposed to the hazard of PDCs are people and infrastruc-tures. A recent example of pyroclastic density currents giving a local impact and damage is the eruption of Merapi (Indonesia) on 2010, which on the vol-cano flanks caused the death of tens of people, among other damages [Jenkins et al., 2013].

1.3 Modeling pyroclastic density currents

Most processes within PDCs are difficult to observe directly, because of their high-destructiveness: measurements are commonly obtained indirectly, from the associated deposits and damages. To better understand PDC propa-gation, some experimental settings on smaller scale have been build [Hogg et al., 1999, Gladstone et al., 2004, Andrews and Manga, 2012] and several analytical and numerical models have been proposed [Sparks, 1976, Valentine and Wohletz, 1989, Dartevelle, 2004, Esposti Ongaro et al., 2007].

Apparently, the most natural way to describe a gas-particle system is by adopting a hybrid approach, combining a continuum description of the gaseous phase with a discrete description for solid grains: in this case the Newton equation is written for each of them. However, if we consider a PDC with solid volume fraction equal to 1% with characteristic grain size equal to 0.5mm we have about 100 millions of particles per cubic meter. This suggests that the discrete approach is not reasonable: even with modern super-computers the computational cost of the problem is too high, since it is necessary to simulate a domain of about 1010 cubic meters. In some cases an hybrid of continuum and discrete approach can be used: only the largest solids, for which a fluid description cannot be used, are simulated directly.

So the most useful approach to describe PDCs is through a continuum description also for solid phase. This can be done or with a single effective pseudo-fluid, or using a complex multiphase description.

In single pseudo-fluid models the equilibrium between gas and particles is assumed. A widely used class of these models adopts the Shallow Water (SW) or depth-averaged approach. The SW models start with the main hy-potheses that in the gravity current the horizontal length scale is much larger

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than the vertical length scale, with the exception of small initial time. Un-der this condition, conservation of mass implies that the vertical velocity of the fluid is small (consequently the vertical velocity component is neglected), moreover the pressure is assumed to be hydrostatic. The motion equations are derived from the Navier–Stokes equations written for the gas-particulate mixture, and averaged along vertical direction (depth-integration) [Sparks, 1976]. These models have been applied to study the scaling laws of particu-late gravity currents [Bonnecaze et al., 1993] and estimate the current runout of PDCs in subcritical (when the typical flow velocity is larger than the wave velocity) and supercritical regimes [Bursik and Woods, 1996]. SW models can describe two-layer currents [Doyle et al., 2010] but the shear stress must be imposed empirically. They are efficient for reproducing natural events if the flow density can be assumed to be constant in time and space: for exam-ple, it was successfully used to reproduce a PDC from the 2006 eruption of Tungurahua volcano [Kelfoun et al., 2009].

Within the same approach, integral models, usually called box model, de-scribing the time-wise evolution of the PDC front and thickness have been developed. This approach extends the work on homogeneous density cur-rents [Benjamin, 1968] to particulate curcur-rents, that can be monodisperse [Hallworth et al., 1998] or polydisperse [Harris et al., 2002]. The decoupling between particles and fluid is modeled thorough a constant settling rate of de-position, and flow properties are assumed to be horizontally uniform. These models have the main advantage of predicting the scaling laws in an analytic or integral form, so that it can be experimentally tested [Gladstone et al., 2004, Hogg et al., 1999]. In [Dade and Huppert, 1996] these results have been applied to the interpretation of the famous Taupo eruption of about 1800 years ago, inverting the box model to estimate the current velocity from the observation of deposits. These models will be the subject of Chap-ter 5.3, where we will discuss the integral approach in more detail.

To develop more realistic simulations, catching the multiphase nature and the vertical stratification typical of PDCs, gas and particle dynamics need to be distinguished and treated separately, adopting the Eulerian-Eulerian approach: the hydrodynamic fields are defined independently for each phase, and the transport equations are written for each of these.

In 1989 Valentine and Wohletz [Valentine and Wohletz, 1989] have used this approach to simulate pyroclastic flows, in two-dimensional domain and with only one particulate class, and other models extended their work to include

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Figure 5: Three dimensional nu-merical simulation, using PDAC code, of an explosive eruption at Vesuvius [Esposti Ongaro et al., 2008] (Rate of mass erupted = 5.0 × 107 _Kg/s). _Spatial

dis-tribution of pyroclasts into the atmosphere, 1000 seconds after the start The isosurfaces repre-sent two different values of par-ticle concentration. The repre-sented domain has dimension of 8km × 8km × 10km.

more particle classes [Dartevelle, 2004].

Only in the last years the first three-dimensional simulations have been performed [Esposti Ongaro et al., 2007], since they require high computa-tional resources. An example of a three dimensional numerical simulation on a real topography is showed in Fig. 5.

However, this approach suffers from our incomplete understanding of the complex physics of PDCs; some models are unable to describe particle de-position and ground erosion, and can only be applied to moderately dilute currents. Moreover in dense currents the interparticle collisions can not be neglected and the solid rheology model plays and important role on current propagation. A different approach to overcame this weakness is given by the kinetic theory of granular flows, a theory developed in last decades to describe the dynamics of particulate systems using the same formalism of kinetic theory of gases, opportunely modified to describe a ”gas” composed by solid particles with inelastic collisions. This approach is widely used in several disciplinary fields, in particular in the engineering context. Dartevelle [Dartevelle, 2004] implemented the results of kinetic theory of granular flows, in which transport equations are derived using the same techniques used in the kinetic gas theory. The main advantage is that the transport parame-ters are expressed as a function of fluidodynamics fields, and the equations

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are fully closed, without additional semi-empirical relations (for example to obtain the dependence of solid viscosity on particle density).

Finally we briefly cite a kinematic model, proposed by Malin and Sheridan (1982), called energy cone, in which they assume that the total mechanical energy decays linearly with the distance. Through a comparison between total energy and potential energy necessary to overcomes an obstacle, this model can be used in hazard assessment studies to estimate the invaded area when initial conditions are known. This model, although it has not a solid theoretical or experimental basis, is often still used in hazard mapping stud-ies, since it provides a fast method to evaluate the effects of the topography on PDC propagation. In the last Chapter we critically analyze the energy cone approach, with a comparison with box-model results.

1.4 Motivation and approach

The main goal of this work is to perform a numerical study on the influ-ence of the vertical stratification of particle concentration on pyroclastic flow propagation. In addition, we are interested to investigate under which con-ditions the flow runout and the longitudinal distribution of the main impact parameters (dynamic pressure and temperature) can be predicted by simpli-fied models.

To achieve these results it is necessary to have a numerical model that correctly describes the key aspects that influence the PDCs. When also the particulate phase is considered as a continuum it is necessary to include in the balance equations for solid mass, momentum and energy the terms that effectively drive the flow dynamics. In particular in the dense basal portion of the flow, where particle-particle collisions play an important role, the rheology of the particulate phases may affect the current stratification.

The resulting model has been implemented in the PDAC numerical model which is able to describe a system composed by a multicomponent gas and N different solid phases, considered as interpenetrating continua. For each particulate phase we have introduced a new hydrodynamic field, the gran-ular temperature. Moreover the dependency of the transport terms on the granular temperatures and other fluid fields is implemented.

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This multiphase model has been applied to perform some numerical ex-periments of the gravitational collapse of a gas-particle mixture in a sur-rounding fluid, and of the horizontally propagating currents (this problem is also know as the dam-break flows). We have analyzed the effects of the initial particle concentration, the grain size and the mixture temperature on vertical stratification in current propagation. The dam-break problem has been thoroughly studied in various regimes and with different methodolo-gies [Benjamin, 1968, Gladstone et al., 2004, Hallworth et al., 1998, Harris et al., 2002]. However, this is the first analysis of the temperature effects in particle-laden currents, in the range of temperature and density contrast typical of pyroclastic density currents.

Simulation results are compared with an integral box model, which pro-vide a simplified analytical description of flow front advancement and the main scaling properties of the phenomenon. This box model is based on several simplifications (e.g. constant volume, constant settling velocity, zero viscosity, small density contrast) and introduce some free parameters, which must be calibrated.

To describe flow regimes typical of pyroclastic density currents we have analyzed the possibility of extending the box model, investigating the ef-fects of different initial conditions, removing some assumptions and propos-ing some modifications more appropriate to describe the dynamics.

Finally, the same dam-break problem is used to start a preliminary study of the effects on flow propagation associated with the considered model for particulate rheology. We compare the evolution of the current position and stratification when we use the equations derived from the kinetic theory with those obtained from the empirical model.

1.5 Outline

In the chapter 2 we start to study multiphase systems, with particular at-tention on those composed by a fluid, that can be a gas or a liquid, and suspended particles. For the Eulerian-Eulerian case we revise the derivation of the fluid equations from the Reynolds transport theorem, introducing also the problem of closure. We then analyze the PDAC model, that adopt a Eulerian-Eulerian approach and has been developed to simulate the multi-phase volcanic flows.

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Chapter 3 is a review of kinetic theory for granular flows. From the study of single particle-particle collisions, the transport equations are derived in a statistical way, using the distribution function of the particle velocities and a Boltzmann-like equation. We focus on the differences with respect to the molecular gas, analyzing the changes induced by the different nature of grain collisions. Moreover we revise the possible critical aspects in hypotheses nec-essary to derive the equations.

In chapter 4 the numerical techniques of PDAC model are briefly ex-plained. Furthermore implementation of the kinetic theory, and the choice for time discretization adopted in new equations are described.

Chapter 5 is dedicated to describe our numerical simulations of the PDCs generated by single column collapse. We examine the characteristics of cur-rent vertical stratification and its influence on curcur-rent dynamics. To in-terpret the simulations results and check the possibility to describing the current propagation through simplified analytical scaling laws we introduce and review the integral box model for particle-driven density currents. Its hypotheses and simplifications are critically revised. The box-model predic-tions are compared with our numerical experiments to investigate when the simplified model well reproduce the front kinematics and when the box model hypotheses are too restrictive to allow a direct transposition to PDCs, and some modifications are needed. The final part of chapter five is dedicated to inspect how the changes in multiphase model derived from the kinetic theory for granular flows influence the current dynamics.

In chapter 6 we show how our results are relevant not only for theoret-ical comprehension of PDCs, but to have direct spillovers also for hazard assessment studies. We present an invasion model which combines the scal-ing laws derived from the box-model, and the idea that the morphology can be accounted for comparing flow kinetic energy with the potential energy associated with a topographic obstacle, as in commonly used energy cone approach. We apply this invasion model to compute the invaded area in a Campi Flegrei caldera, for different initial conditions.

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2 Modeling multiphase flows

We consider as multiphase flow any fluid flow consisting of more than one phase or component Multiphase flows can be found everywhere in nature (e.g. snow, fog, avalanches, mud slides, sediment transport, debris flows) and are also important for industrial applications, such as chemical, petroleum, and power generation industries.

We here describe the PDAC model, that uses an Eulerian-Eulerian ap-proach to describe a system composed by one multicomponent gaseous phase and several different solid phases, considered as interpenetrating continua. For each phase the balance equation for mass, momentum and energy are written.

2.1 Overview of gas-solid flow balance equation

There are two main approaches to model fluid-solid flows. The first, called Eulerian-Lagrangian, tracks the motion of each particle and solve the dynam-ics of the fluid at a length scale larger than the particle diameter (microscopic length scale). On the other hand, the Eulerian-Eulerian models treat the fluid and solid phases as interpenetrating continua, and study their dynamics by means of averaged equations of motion. The second approach is often the only possible because it is computationally less expensive.

In this work the Eulerian-Eulerian description is adopted, where solid phases are considered as a continua. To do this, it is assumed that the system consists of a sufficient number of particles so that macroscopic properties, and their derivatives, exist and are continuous (a critical analysis of this assumption will be done in chapter 3).

For a control volume V (t), which may change in time, we consider a generic intensive fluid property ψ. The rate of variation of the integral of ψ over the control volume V (t) is

d dt

Z

V (t)

ψdV (2.1)

and since the volume depends on time we can not exchange the integral and the time derivative.

This integral can be rewritten by applying the Reynolds transport theo-rem (e.g. see [Gidaspow, 1994]). If the control volume moves with velocity

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v, d dt Z V (t) ΨdV = Z V (t) ∂tΨ + ∇ · Ψv dV (2.2)

where the velocity of the control volume is it is assumed well defined for each point in V (t), and the divergence theorem is used.

Generally we refer to multicomponent flow when the fluid is composed by several species which are well mixed, such as a gaseous mixture, in this case the volume V can be assumed to be the same for all species. On the other hands, if we can clearly identify a separation between the phases or the components of the fluid we call the system as multiphase flow (examples of multiphase flow are those composed by gas or liquid and macroscopic solid particles, or by two different immiscible liquids as oil and water). In this case the volume occupied by a given phase cannot be occupied at the same tame by other phases. In multiphase flow, each of the phases is considered to have a separately defined volume fraction i (the sum of which is unity),

from which the total volume associated with the phase can be computed,

Vi =

Z

V (t)

idV (2.3)

The mass of i-th phase depends on its density, ρi, and volume fraction,

mi =

Z

V (t)

ρiidV (2.4)

and its balance equation can be obtained from Reynolds theorem, setting ψ = ρii. If there are no phase change, and if we consider a control volume

which moves with the same phase velocity vi (in multiphase model for each

phase we define a fluid velocity), mi is preserved and the left side of (2.2)

vanishes. Noting that the control volume can be chosen arbitrarily small, it is possible to rewrite the Eq. (2.2) in a differential form,

∂ρii

∂t + ∇ · (ρiivi) = 0 (2.5) which gives the continuity equation for the phase i.

Similarly the momentum associated with the phase i is defined as Z

V (t)

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and its rate of change can be obtained equating the right side of (2.2) with the total force acting on the considered phase, f_i∗,

d dt

Z

V (t)

ρiividV = fi∗ (2.7)

The total force f_i∗ is usually decomposed as a sum of the external contri-bution fext, and the contact forces contribution, expressed as the divergence of the stress tensor, ∇Pi. Additionally, in multiphase flow, an interphase

force fi,j is usually considered, [Gidaspow, 1994],

∂(iρivi) ∂t + ∇ · (iρivivi) = ∇Pi+ f ext +X j6=i fi,j (2.8)

with the constrain on interphase pair force fj,i = −fi,j (Newton

action-reaction law).

Eq.(2.8) is not closed, since it requires the knowledge of the terms on the right hand side. The simplest expression for the stress tensor Pi, is

proportional-ity to the identproportional-ity matrix, through the definition of a phase pressure Pi, P =

−PiI, holding for inviscid fluids. In other models, which hold also for viscous

flows, the stress tensor assumes a Newtonian form (generalization of Navier-Stokes equation for a multiphase system), Pi = −P I + µ [∇vi+ (∇vi)t], or

even more complex dependency on fluid fields.

In single phase fluid it is customary to write an energy balance for the internal energy. Similarly also for the multiphase flow an energy balance equation for the internal energy Ei of the phase i, defined as,

Ei =

Z

V

ρiiei (2.9)

where ei is the density of ith internal energy for unit mass. However, in

multiphase flow some unexpected work terms arise, and a simpler approaches is often used [Gidaspow, 1994], used also in elementary thermodynamics. The resulting balance equation for internal energy Ei has the same form of

single-phase flow, with the (eventual) addition of the rate of heat exchange between phase i and j, ˜qi,j, and a term involving work of expansion of void fraction,

proportional to the pressure (defined as the diagonal coefficient of the stress tensor), Pi∂ti, ∂(iρiei) ∂t + ∇ · (iρieivi) = −∇ · (qi) − Pi∂ti− Pi∇ivi+ X j6=i ˜ qi,j (2.10)

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where qi is the energy flux, which in the simplest form is proportional to the

gradient of the temperature.

The term Pi∂ti can be eliminated writing the rate change of enthalpy, hi ≡

(ei+ Pi/ρi). Combining the equations (2.5), (2.10), with the differential form

of enthalpy increase, dhi ≡ dei+ ρi−1dPi+ Pidρi−1 (2.11) we obtain ∂(iρihi) ∂t + ∇ · (iρihivi) = −iTi : ∇vi+ i ∂Pi ∂t + vi· ∇Pi − ∇ · qi+ X j6=i ˜ qi,j (2.12) where (Ti) is the non-diagonal part of stress tensor, Pi = P I + Ti. Both

entropy and energy representation require the expressions of the heat flux and the interphase exchange terms to be in a closed form.

2.2 The PDAC model

In this section we analyze the PDAC (Pyroclastic Dispersal Analysis Code) code (see [Esposti Ongaro et al., 2007]), implemented to simulate the disper-sal dynamics of pyroclasts in the atmosphere during explosive eruptions.

This code describes the injection and dispersal of a hot and high-velocity gas-pyroclast mixture in a steady standard atmosphere. The gas phase may be composed of different chemical components leaving the crater, such as water vapor and carbon dioxide, and atmospheric air, considered as a single chemical component. The pyroclasts are described by N phases of solid par-ticles, each one characterized by a diameter, density, specific heat, thermal conductivity and viscosity.

The main hypotheses in PDAC model are:

• the solid particles and the gas are considered as interpenetrating con-tinua (Euler-Euler approach);

• particles fragmentation and aggregation are neglected;

• sedimentation and erosion of the solid particles on the ground is not accounted for;

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• no phase changes or chemical reactions are considered;

• the heat transfer between different solid phases is considered propor-tional to the spatial derivative of temperature. Similarly the viscous stress tensor is considered proportional to velocity gradient.

• the gas is compressible, solid particles are incompressible; • we impose the ideal gas law for gaseous phase;

• turbulence is accounted for in gas stress tensor, by an effective turbu-lence viscosity.

For each phase a set of five scalar equations is written, giving the balance of mass, momentum components and enthalpy, expressed by the equations (2.5), (2.8), and (2.12). The subscript i = g indicates the gaseous phase, i = 1, . . . , N the solid phases.

Additionally the gas phase can be composed by M different molecular species. For each of them the balance equation for the mass fraction yi is

written,

∂

∂t(gρgyi) + ∇ · (gρgyivg) = 0, i = 1, 2, ..., M. (2.13)

2.2.1 Constitutive equations

We have a set of 5N + (M − 1) coupled partial differential equations. To solve this system of equations some constitutive relations are required. These equations express equation of state of gas, stress tensors for each phase, gas-particle and gas-particle-gas-particle drag terms, gas-gas-particle thermal conductivities and gas-particle heat fluxes as a function of fluids fields.

Volumetric and mass fraction closure The gas components fraction and the single-phase volumetric fraction are not independent, since the con-straints must be verifed

X i yi = 1, X i i = 1 (2.14)

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Equation of state The gas density depends on pressure and temperature throughout the equation of state of ideal gases,

ρg(T ) =

P

e

RT (2.15)

and the temperature can be computed from the enthalpy,

Tg = hg CP,g Ts = hs CP,s (2.16)

where CP,i indicate the specific heat at constant pressure for unit mass. For

the gas phase CP,i depends on the chemical composition,

CP,i =

X

i

yiCP,i (2.17)

Gas stress tensor The gas phase stress tensor is modeled by adopting a turbulent subgrid scale model. In the PDAC model the large eddy simulation (LES) approach is followed. The filter used is the box-filter. This operation gives rise to an extra turbulent term, the subgrid turbulent dissipation, which is described by the introduction of an eddy turbulent viscosity as proposed in the Smagorinsky model [Smagorinsky, 1963]. The gas phase stress tensor takes the following form:

Pg = µgτeg+ µgtτg (2.18) with τg = h ∇vg+ (∇vg)T i , τ_eg = τg − 2 3(∇ · vg)I (2.19) and µgtis an effective viscosity expressed as the sum of the gas shear viscosity

(µg) and Smagorinsky viscosity.

Solid stress tensor The stress tensor of solid s-th phase is described in terms of a Newtonian viscous component and a Coulombic repulsive compo-nent,

Ps= µsτev,s− τc,sI, s = 1, 2, ..., N (2.20) where the viscous tensor is

e τv,s = h ∇vs+ (∇vs)T − 2₃(∇ · vs)I i (2.21)

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and the Coulombic component gradient is defined by

∇τc,s = G(s)∇s. G(s) = 10−as+b (2.22)

These relations are obtained from semi-empirical studies [Gidaspow, 1994]. The Coulombic repulsive factor is introduced to prevent unrealistically high particle concentration. Solid viscosity is assumed proportional to the volume fraction,

µs = css(0.5 ≤ cs ≤ 2) (2.23)

whose validity is established experimentally up to concentration of a few per-cent [Miller and Gidaspow, 1992] (see Fig. 6).

Interphase force Gas and solids momentum equations are coupled through a drag force, proportional to the difference in velocity fields,

fg,s= Dg,s(vs− vg) (2.24)

Gas-particle drag coefficients are derived from semi-empirical studies, their validity may depend on the flow regime, thus including also correction for turbulence and different particle concentrations. In the dilute regime, for g ≥ 0.8, we adopt the drag expression

Dg,s = 3 4Cd,s gsρg | vg− vs| ds −2.7_g , s = 1, 2, ..., N (2.25) where Cd,s = 24 Res [1 + 0.15 Re0.687_s ], Res< 1000 (2.26) = 0.44, Res ≥ 1000, (2.27)

For g < 0.8, Dg,s is given by:

Dg,s = 150 2_sµg gd2s + 1.75sρg | vg − vs | ds , s = 1, 2, ..., N (2.28)

where Res is the particles Reynolds numbers,

Res =

gρgds| vg− vs|

µg

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The interaction between different particulate phases are described by analogous particle-particle drag force, proportional to vi − vj, with a drag

coefficient expressed as

Ds,p= Fspα(1 + e)ρssρpp

(ds+ dp)2

(ρsd3s+ ρpd3p)

| vs− vp |, p 6= s. (2.30)

This depends on collision restitution coefficient, e ≤ 1, and is a function of the two volume fractions, Fsp [Esposti Ongaro et al., 2007].

Heat flux For the heat fluxes the Fourier’s law is used,

qi = −kii∇Ti (2.31)

when the effective conductivities {ki} take into account for turbulence on

unresolved length scales.

The rate of heat transfer between the gas and the solid phases is given by the product of a transfer coefficient Qs and a driving force, which is the

dif-ference of temperature between the two phases. The coefficient Qsrepresents

the volumetric interphase heat transfer coefficient which equals the product of the specific exchange area and the fluid-particle heat transfer coefficient expressed in terms of an empirical expression for the Nusselt number N us,

Qs = N us 6sgs d2 s , s = 1, 2, ..., N (2.32) N us = (2 + 52s)(1 + 0.7Re 0.2 s P r 1/3_{) + (0.13 + 1.2}2 s)Re 0.7 s P r 1/3_, _(2.33)

when P r = (Cpgµg)/(sg) is the Prandl number and kg is the thermal

con-ductivity. The rate of heat exchange between different particulate phases are not considered in PDAC model.

2.3 Critical aspects of closure equations

As we have seen, the transport equations require some additional closure equations. Using a continuum approach, these closure relations can be set experimentally, or determined using a semi-empirical approach to match the numerical result with the observations. This approach exhibits two main weakness:

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Figure 6: Dependency of the effective solid viscosity on the solid volumetric fraction. Figure exctracted from [Miller and Gidaspow, 1992].

• The rheological models thus obtained are often non universal, but de-pend strictly on fluid condition (e.g. particle size, particle concentra-tion, fluid velocity, Reynolds number)

• Often the range of applicability of these rheological models is not suf-ficiently clear.

Miller and Gidaspow (1992) investigated in laboratory the effective solid viscosity and its dependency on the solid volumetric fraction. Their results are showed in Fig.6, and the linear regression µs = cs is a good

approxima-tion only for solid concentraapproxima-tion less than few percent.

For these reasons in the next chapter we review the kinetic description of granular flows, in which all terms in fluid equations depend only on the fluid fields and particle properties, and can account for a wide range of concentra-tions.

Moreover, a better definition of the limits of applicability is immediately provided when the theoretical hypotheses are understood. The importance of the derivation of fluid equations from kinetic description goes beyond the

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knowledge of a single transport coefficient but has a relevance to better un-derstand the complex aspects of multiphase flows.

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3 Review of kinetic theory of granular flows

In this chapter we present an overview of the kinetic theory for granular flows, from the theoretical assumption to the derivation of the fluid equations and of the transport coefficients.

Firstly, in section 3.1, the granular flows are briefly described, and the particle collisions are studied, introducing the idea of an inelastic restitution coefficient to take into account the kinetic energy dissipation.

In the next two sections we revisit the derivation of the Boltzmann equa-tion for granular system, starting from the Liouville theorem. From the Boltzmann equation we find the transport equations for mass density, mo-mentum and energy, which require the knowledge of the distribution function in a closed form.

In section 3.4 one approximate method of solution of Boltzmann equa-tion is showed, following the work of Chapman and Enskog (1961) for dense molecular gases out of equilibrium, and we give the resulting terms in hydro-dynamic equations.

Finally, in the last section, we show the complications that occur when we extend the kinetic description to multiphase granular system. We concisely rerview the models that can be found in literature.

3.1 Granular flows

Granular materials are large conglomerations of discrete macroscopic solid particles, with or without an interstitial fluid. These systems exhibit many characteristics, which cannot be found in ordinary fluids such as air and wa-ter, and solids, such as metal. Like solids they can sustain shear stress at rest, and can assume an equilibrium state in which the free surface can be inclined with a certain angle over the horizontal. Like liquids, they flow from vessels under the action of gravity, but the mass flow rate is approximately independent of the height of material above the discharge orifice [Aguirre et al., 2011]. Unlike water, granular materials are compressible in the sense that the space between the particles often changes during flow, but the total volume can not be smaller than a threshold value, corresponding to max-imum package. This maxmax-imum granular fraction depends on the particle shape, for spherical particles it is about 0.63.

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1996], depending on the average fluctuating kinetic energy per grain. When the individual particles are fairly at rest relatively to each other, and the fluctuating energy is irrelevant, the system is named as granular solid. On the other hand, when the relative motion becomes important, we have a granular flows. Moreover granular flows can be subdivided into granular liquids and granular gases. In the first case the interactions between particles are frictional and can be mobilized to different degrees depending on the preparation history, giving rise to memory effects [Arnarson and Jenkins, 2004]. Whereas, in the second case, the particles are in continuous motion and contacts between the grains are almost instantaneous. We focus our analysis on granular gases, but we can’t forget this peculiarity of granular flows, since the same system can exhibit a different behavior depending on flow conditions (i.e., flow velocity, particle concentration) and different flows regimes can coexist [Bareschino et al., 2008, Larcher and Jenkins, 2013].

The evolution of granular gas is completely determined by pure mechan-ical collisions of its particles, and by the external force, if present. Therefore a theoretical description of granular gases requires detailed understanding of the mechanism of collisions between particles.

3.1.1 Dynamics of binary collisions

The grain collisions in granular flows do not preserve the total kinetic energy. The inelasticity of collisions is a result of plastic deformation occurring within the particles during collision, and consequent dissipation of kinetic energy into thermal heat. This dissipated energy depends on particle properties (e.g., material, size, density) and the relative impact velocity. However in most studies a simplified model of grain collisions is used [Jenkins and Savage, 1983], whose main assumptions are:

• a collision occurs when the distance between the particles is exactly zero

• particles are considered as smooth hard spheres, without any internal degree of freedom

• we consider identical particles of mass m

• the momentum components orthogonal to the line joining the particle centers remains unchanged during collision

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Figure 7: Sketch of collision be-tween two particles with diameter dp, initial velocities c1and c and

fi-nal velocities c0₁ and c0.

• inelasticity is taken into account with a single coefficient that does not depend on impact velocity

With these hypothesis we can write

(k · c0₁₂) = −ep(k · c12), (3.1)

where c12 and c012are the translation velocity of particle 1 relative to particle

2 before and after the collision, k is the unit vector from center of particle 1 to center of particle 2, and ep is the restitution coefficient.

If two particles have initial velocities respectively c1 and c2before impact,

and if we identify the velocity components along the k-direction with u1 and

u2 we can rewrite the equation (3.1) as

u0₁− u0₂ = −ep(u1− u2). (3.2)

Combining last equation with conservation of momentum component parallel to k,

u0₁+ u0₂ = u1 + u2 (3.3)

we obtain the final particle velocities,

u0₁ = u1+ 1 + ep 2 (u2− u1) ≡ u2,1, u 0 2 = u2− 1 + ep 2 u2,1, (3.4) and finally we have the expression for energy loss during collision,

∆E = m 2(u 02 1 + u 02 2 − u 2 1− u 2 2) = − m 4(1 − e 2 p)u 2 21, (3.5)

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this energy variation is proportional to (1 − e2

p), when ep is equal to 1 the

energy is preserved. The opposite limit, with maximum dissipation, corre-sponds to ep → 0.

In some cases, for granular gases, the assumption of quasi-elastic collisions is done, also for our purpose we can consider only this case, in agreement with experimental observations. Since the kinetic energy is not conserved the particles can not remains in granular gas regime for a long time without an external energy source. In more general case, when colliding particles have different mass and size, we can derive the energy loss with the only change in conservation of momentum (3.3),

, m1u01+ m2u20 = m1u1+ m2u2 (3.6)

where m1 and m2 are the mass of the colliding particles. Consequently the

dissipated energy is given by

∆E = − m1m2 2(m1+ m2)

(1 − e2_p)u2₂₁. (3.7)

In the next sections we will discuss the importance and the consequence of an inelastic factor ep < 1, and the necessary hypotheses to derive the fluid

equation in a rigorous way. The inelasticity of collisions giving rise to some of granular flows peculiarities, for example grain clustering and collapsing, and only in the limit e ∼ 1 we can reach a quasi-equilibrium state.

We considered here the simplest case of particle-particle collision, from which is possible to derive the macroscopic laws adopting the methodology of the statistical mechanics. In addition, there are more complicated kinetic theory for granular flows, which consider particles with different mass and size [Jenkins and Mancini, 1989, Arnarson and Jenkins, 2004, Larcher and Jenkins, 2013], highly dense flows, in which particle friction can not be ne-glected [Johnson and Jackson, 1987, Jenkins, 2007], and more complicated models for energy dissipation [Brilliantov and P¨oschel, 2010] (i.e., introducing a dependence of restitution coefficient on colliding velocity).

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3.2 Derivation of Boltzmann equation

3.2.1 Liouville equation and BBGKY hierarchy

We consider a closed system that consist of N identical classical particles. The position and velocity coordinates of the fluid particles will be denoted by {ri} and {ui}, and then the state of the system at a generic time t is

completely characterized by the positions and velocities of all particles at that time, and it can be represented by a point

(r1, c1, . . . , ri, ci, . . . , rN, cN) ≡ (Ω1, . . . , ΩN) ≡ Ω (3.8)

in a 6N dimensional phase space. In principle it is possible to solve the equation of motion for each particles from initial conditions, integrating the Newton’s equation, but in practice this approach is unrealizable, since in general we do not know the initial values of position and velocity for each particles and since the computational cost will be too high to simulate a long time.

For this reason we adopt a statistical approach by introducing the N -particle distribution function fN in a 6N -dimensional space,

dW = fN(Ω, t)dΩ (3.9)

where dW is the probability to find the mentioned system at time t in the element dΩ of phase space, and the normalization of fN is fixed by the total

probability

Z

Ω

fN(Ω, t)dΩ = 1 (3.10)

When the particle dynamics can be described in terms of an Hamilto-nian function (i.e., the energy is preserved) the Liouville equation gives the evolution of the N -particle distribution function,

∂fN ∂t + N X i=1 vi· ∂fN ∂ri + N X i=1 Fi· ∂fN ∂vi = 0 (3.11)

where vi is the velocity of a particle numbered i, Fi is the force on unit mass

acting on the ith particle. It is worth noting that such an equation has not a theoretical justification for a system with energy loss, although Liouville equation is commonly used in the kinetic description of granular flows.

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Essentially the equation (3.11) expresses that the distribution function fN does not change along the trajectory into the phase space:

fN(Ω(t0), t0) = fN(Ω(t), t) (3.12)

where Ω(t0) is the point of phase space representative of the system at time t0 corresponding to initial condition Ω(t) at time t.

Eq.(3.11) has 6N variables, so a complexity reduction is required, to obtain a feasible set of equations. To do this we can also introduce an s-particle distribution function, s < N , in accordance with the definition:

fs =

Z

fN(Ω1, . . . , ΩN, t)dΩs+1. . . dΩN (3.13)

and integrating the equation (3.11) with respect dΩs+1. . . dΩN we obtain the

equation that describes the evolution of the s-particle distribution function,

Z _∂f N ∂t + N X i=1 vi· ∂fN ∂ri + N X i=1 Fi· ∂fN ∂vi ! dΩs+1. . . dΩN = 0 (3.14)

Consider the first term in last integral, we can invert the order of integration and differentiation because the limits of integration do not depend on time,

Z _∂f

N

∂t dΩs+1. . . dΩN = ∂fs

∂t (3.15)

Precisely in the same manner we change the order of operations in the second term for i ≤ s, putting out of the integrals the spatial derivatives. For all i > s in can be shown that the integral is identically zero, using Gauss the-orem and the property that distribution function should tend to zero, when |ci| goes to infinity.

Denoting the external force acting on the i-th particle as Fe

i we can rewrite

the force term,

Fi = Fei + N

X

j=1

Fi,j (3.16)

where Fi,j(ri, rj) is the force acting on the ith particle from the particle j.

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the same arguments used previously for the second term are valid, and the sum can be restricted up to s. Substituting (3.16), and with little algebra, the evolution equation for fs is finally given by

∂fs ∂t + s X i=1 ci· ∂fs ∂ri + s X i=1 Fe_i · ∂fs ∂ci + s X i,j=1 Fi,j· ∂fs ∂ci = − s X i=1 (N − s) ∂ ∂ci · Z fs+1(Ω1, . . . , Ωs+1, t)Fi,s+1dΩs+1 (3.17)

that is usually known as BBGKY hierarchy (Bogoliubov–Born–Green–Kirkwood–Yvon hierarchy, sometimes called Bogoliubov hierarchy) [Kirkwood, 1946].

The first two terms of BBGKY hierarchy gives the dynamics of one-particle distribution function,

∂f1 ∂t + c1· ∂f1 ∂r1 + Fe₁· ∂f1 ∂c1 = (1 − N ) ∂ ∂c1 · Z f2F1,2dΩ2 (3.18)

and of the pair distribution function,

∂f2 ∂t + c1· ∂f2 ∂r1 + c2· ∂f2 ∂r2 + Fe₁· ∂fs ∂c1 + Fe₂· ∂fs ∂c2 + F1,2· ∂fs ∂c1 + F2,1· ∂fs ∂c2 = (2 − N ) ∂ ∂c1 · Z f3F1,3dΩ3+ ∂ ∂c2 · Z f3F2,3dΩ3 ! (3.19)

The set of integro-differential equations turns out to be a coupled one, so that the distribution function fs depends on fs+1. At first sight the solution

procedure for such a set should be as follows. First find the distribution function fN and then solve the set of BBGKY equations subsequently for

de-creasingly lower-order distributions. But if we know the function fN, there is

no need at all to solve the equations for fs, since it can be computed directly

for fN from definition of fs. This suggests that the rigorous solution to the

set of BBGKY equations is again equivalent to solving Liouville equations, but using some assumptions it is possible to derive some simplified equations for lower orders.

Consider now a gas of N identical hard spheres of radius dp, to simplify the

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radius, that will be useful in the extension to systems composed by solid grains. The interacting potential between two different particle is function of the distance between centers, ri,j,

Ui,j(rij) =

∞, when rij ≤ dp

0, when rij > dp

(3.20)

and the post-collisional velocities depend on pre-collisional velocities as is explained in previous paragraph, with elastic factor ep = 1. Since different

particles cannot overlap, it must be verified that

fs(r1, c1, . . . , ri, ci, . . . , rj, cj, . . . , rs, cs, t) = 0 if rij ≤ dp (3.21)

and combining this argument with the constrain Fij = 0 for ri,j > 0, we see

that in right hand term of eq. (3.14) the only non-zero contribute to integral is when two particles are distant exactly dp.

Inserting the collision rules, and performing the integral on spatial vari-able, we can finally obtain the BBGKY hierarchy for hard sphere gas [Puglisi et al., 1999] ∂fs ∂t + s X i=1 ci· ∂fs ∂ri + s X i=1 Fe_i · ∂fs ∂ci + s X i,j=1 Fi,j· ∂fs ∂ci = d2_p s X i=1 (N − s) Z k·c12>0 f0 s+1− fs+1c12· kdkdv2 (3.22)

where f_s+10 is the distribution function of post-collisional in terms of the pre-collisional ones.

3.2.2 Boltzmann-Grad limit

To derive the Boltzmann equation of the kinetic theory of gases from N -body problem of classical mechanics we take in each term of BBGKY hierarchy the limit N → ∞. The right hand side of equation (3.22) essentially gives the variation of s-particle distribution function on a characteristic time between collisions, ∆fs/τcoll. Since τcoll ∼ N d2p to keep it finite when N → ∞ one

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To obtain the Boltzmann equation it is necessary the assumption of molec-ular chaos, that express each pair particle distribution function in factorized form

f2(r1, c1, r2, c2, t) = f1(r1, c1, t)f1(r2, c2, t) (3.23)

At an intuitive level, one can realize that, in a very dilute gas, when two particles collide it is very unlikely that they have met before, either directly or indirectly through a common set of other particles, so their velocities can be considered as uncorrelated. It is important to highlight that the molecular chaos assumption is possible only for particles that are about to collide; after the collision, the scattered velocities are strongly correlated. In B-G limit we consider the particles as a point, and then the location of colliding particles in BBGKY hierarchy can be set to an unique position in space, r. Moreover, in the same limit the multiplicative factor (N −s) is approximately N for fixed s.

With these assumptions we can rewrite the evolution equation for one particle distribution function, omitting the subscript 1 on f and the super-script e on external force for simplicity:

∂f ∂t+ c · ∂f ∂r+ F · ∂f ∂c = N d 2 p Z k·c12>0 f (c0 )f (c0₂) − f (c)f (c2)c12· kdkdc2 (3.24)

where c0, c0₂ are particle post-collisional velocities, and their dependence on the colliding velocities is through the rules show in section 3.1.1. This equa-tion is the Boltzmann equaequa-tion for hard spheres gas. With substitution k → −k, where the distribution function is isotropic, we can substitute the integral over directions which satisfy the constrain c12· k with the half of the

integral over all directions in space,

∂f ∂t + c · ∂f ∂r + F · ∂f ∂c = Q(f, f ) (3.25) Q(f, f ) ≡ N d 2 p 2 Z f (c0 )f (c0₂) − f (c)f (c2)c12· kdkdc2 (3.26)

It is important to emphasize that in the derivation of Boltzmann equa-tion not only the factorizaequa-tion of pair distribuequa-tion funcequa-tion is assumed, but also the substitution of colliding distribution function with the asymptotic f ,

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before and after particle interaction. This approximation is reasonable when the particle diameter dp, and the characteristic interaction time τi = dp/v0

(v0is the typical particle velocity), are much smaller than the mean free path

l, and the characteristic time scale associated, the collision time τc = l/vth,

so that the collision can be considered instantaneous and local. Moreover, when this hypothesis is verified, the simultaneous collisions between three or more particles are improbable and we can consider only binary events.

In addition, in an inhomogeneous gas, one can introduce a characteristic hydrodynamic length L, which is the typical range over which the distribution function changes appreciably. Since the asymptotic f is used, it is required that the distribution function must be approximately uniform on a length scale typical of particle interaction, that implies l L. Equivalently the hydrodynamic time, τL = L/v0, must be much bigger than the mean free

time τl.

3.2.3 Boltzmann equation for granular gases

From a simple estimation of the mean free path can be shown that the con-dition rd l is equivalent with nd3p ≡ s 1, so that the mean volume

occupied by the particles must be small (dilute regime).

For a dense gas, in which the particle size cannot be neglected, the Boltz-mann equation is not expected to be appropriate. On the other hand, Enskog proposed in 1922 [Chapman and Cowling, 1970] a heuristic modification of the Boltzmann equation for a system of hard spheres accounting for finite-density effects. First, the Enskog equation takes into account that the centers of the two colliding spheres are separated by a distance equal to the diam-eter. This incorporates the transport of momentum and energy occurring across finite distances in collisions, which is important in dense gases and liquids. Second, the collision frequency is increased by a factor χ associated with the spatial correlations due to excluded volume effects. Otherwise, the assumption of molecular chaos is maintained.

f2(r, c, r + dpk, c2, t) = χ(r + ₂1dpk)f (r, c, t)f (r + dpk, c2, t) (3.27)

In literature several different expressions of χ are proposed, each of these must assume value 1 when density is zero and must diverge into the opposite limit, when particle fraction goes to maximum packing fraction permitted

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by particles shape (about 0.63 for identical spheres). We have adopted the model proposed by [Ogawa et al., 1980],

χ r + dp

2 k =

h

1 − (s/maxs )

1/3i−1_, _(3.28)

where s is the local volume fraction, that can assume a maximum value

equal to max_s .

With these approximations the collision term in Boltzmann equation gets

Q(f, f ) = N d 2 p 2 Z χ(r + 1 2dpk)f (r, c 0 )f (r − dpk, c02)− χ(r − 1₂dpk)f (r, c, t)f (r − dpk, c2, t)c12· kdkdv2 (3.29)

Let us consider a granular gas composed of smooth inelastic hard spheres with collision rules as expressed in paragraph 3.1.1 and restitution coefficient ep . The free streaming evolution of distribution function is well described

by the left hand side of Boltzmann equation (3.25), but to take into account the inelasticity we must modify the collision integral [Jenkins and Savage, 1983, Farrell et al., 1986].

In fact the integral (3.29) is a balance of direct encounters, in which one particle has pre-collisional velocity equal to c, and inverse encounters, where one scattered particle has final velocity c. In the inelastic case the inverse collision frequency is multiplied by a factor e−2_p , that descend by the product k0· c0

12 and the Jacobian associated with the transformation from

the pre-collisional grain velocities to the post-collisional ones. Consequently the Boltzmann equation assumes the form

∂f ∂t + c · ∂f ∂r + F · ∂f ∂c = N d2 p 2 Z h 1 ep2 χ(r +1₂dpk)f (r, c0)f (r − dpk, c02)− χ(r − 1₂dpk)f (r, c, t)f (r − dpk, c2, t) i c12· kdkdv2 (3.30)

Multiphase flow modeling and numerical simulation of pyroclastic density currents

Universit`

a di Pisa

Facolt`

a di Scienze Matematiche, Fisiche e Naturali

School of Graduate Studies “Galileo Galilei”

Ph.D Thesis in Applied Physics

Multiphase flow modeling and

numerical simulation of

pyroclastic density currents

Contents

List of Tables

List of Figures

List of Symbols

Acknowledgements

1

Introduction

1.1

Mechanism of explosive eruptions

1.2

Phenomenology of PDCs

1.3

Modeling pyroclastic density currents

1.4

Motivation and approach

1.5

Outline

2

Modeling multiphase flows

2.1

Overview of gas-solid flow balance equation

2.2

The PDAC model

2.3

Critical aspects of closure equations

3

Review of kinetic theory of granular flows

3.1

Granular flows

3.2

Derivation of Boltzmann equation